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Theorem sge0tsms 42204
Description: Σ^ applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
sge0tsms.g 𝐺 = (ℝ*𝑠s (0[,]+∞))
sge0tsms.x (𝜑𝑋𝑉)
sge0tsms.f (𝜑𝐹:𝑋⟶(0[,]+∞))
Assertion
Ref Expression
sge0tsms (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))

Proof of Theorem sge0tsms
Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2795 . . . 4 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )
21a1i 11 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
3 xrltso 12384 . . . . . 6 < Or ℝ*
43supex 8773 . . . . 5 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V
54a1i 11 . . . 4 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V)
6 elsng 4486 . . . 4 (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
75, 6syl 17 . . 3 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
82, 7mpbird 258 . 2 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
9 sge0tsms.x . . . . . . 7 (𝜑𝑋𝑉)
109adantr 481 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
11 sge0tsms.f . . . . . . 7 (𝜑𝐹:𝑋⟶(0[,]+∞))
1211adantr 481 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13 simpr 485 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 42197 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
1511ffnd 6383 . . . . . . . . 9 (𝜑𝐹 Fn 𝑋)
1615adantr 481 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
17 fvelrnb 6594 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1816, 17syl 17 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1913, 18mpbid 233 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
20 iccssxr 12669 . . . . . . . . . . . . . 14 (0[,]+∞) ⊆ ℝ*
21 sge0tsms.g . . . . . . . . . . . . . . 15 𝐺 = (ℝ*𝑠s (0[,]+∞))
22 simpr 485 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
2311adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
24 elinel1 4093 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
25 elpwi 4463 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2624, 25syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
2726adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
28 fssres 6412 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥𝑋) → (𝐹𝑥):𝑥⟶(0[,]+∞))
2923, 27, 28syl2anc 584 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
30 elinel2 4094 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3130adantl 482 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
32 0red 10490 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
3329, 31, 32fdmfifsupp 8689 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) finSupp 0)
3421, 22, 29, 33gsumge0cl 42195 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ (0[,]+∞))
3520, 34sseldi 3887 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
3635ralrimiva 3149 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
37363ad2ant1 1126 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
38 eqid 2795 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))
3938rnmptss 6749 . . . . . . . . . . 11 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
4037, 39syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
41 snelpwi 5228 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
42 snfi 8442 . . . . . . . . . . . . . . 15 {𝑦} ∈ Fin
4342a1i 11 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ Fin)
4441, 43elind 4092 . . . . . . . . . . . . 13 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
45443ad2ant2 1127 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
4611adantr 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → 𝐹:𝑋⟶(0[,]+∞))
47 snssi 4648 . . . . . . . . . . . . . . . . . . 19 (𝑦𝑋 → {𝑦} ⊆ 𝑋)
4847adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → {𝑦} ⊆ 𝑋)
4946, 48fssresd 6413 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
5049feqmptd 6601 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
51 fvres 6557 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹𝑥))
5251mpteq2ia 5051 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))
5352a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5450, 53eqtrd 2831 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5554oveq2d 7032 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
56553adant3 1125 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
57 xrge0cmn 20269 . . . . . . . . . . . . . . . . 17 (ℝ*𝑠s (0[,]+∞)) ∈ CMnd
5821, 57eqeltri 2879 . . . . . . . . . . . . . . . 16 𝐺 ∈ CMnd
59 cmnmnd 18648 . . . . . . . . . . . . . . . 16 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
6058, 59ax-mp 5 . . . . . . . . . . . . . . 15 𝐺 ∈ Mnd
6160a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐺 ∈ Mnd)
62 simp2 1130 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
6311ffvelrnda 6716 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ (0[,]+∞))
64633adant3 1125 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ (0[,]+∞))
65 df-ss 3874 . . . . . . . . . . . . . . . . . 18 ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
6620, 65mpbi 231 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
6766eqcomi 2804 . . . . . . . . . . . . . . . 16 (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
68 ovex 7048 . . . . . . . . . . . . . . . . 17 (0[,]+∞) ∈ V
69 xrsbas 20243 . . . . . . . . . . . . . . . . . 18 * = (Base‘ℝ*𝑠)
7021, 69ressbas 16383 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
7168, 70ax-mp 5 . . . . . . . . . . . . . . . 16 ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
7267, 71eqtri 2819 . . . . . . . . . . . . . . 15 (0[,]+∞) = (Base‘𝐺)
73 fveq2 6538 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
7472, 73gsumsn 18794 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑦𝑋 ∧ (𝐹𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
7561, 62, 64, 74syl3anc 1364 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
76 simp3 1131 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
7756, 75, 763eqtrrd 2836 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
78 reseq2 5729 . . . . . . . . . . . . . 14 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
7978oveq2d 7032 . . . . . . . . . . . . 13 (𝑥 = {𝑦} → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
8079rspceeqv 3577 . . . . . . . . . . . 12 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
8145, 77, 80syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
82 pnfxr 10541 . . . . . . . . . . . . 13 +∞ ∈ ℝ*
8382a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ℝ*)
8438elrnmpt 5710 . . . . . . . . . . . 12 (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8583, 84syl 17 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8681, 85mpbird 258 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
87 supxrpnf 12561 . . . . . . . . . 10 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
8840, 86, 87syl2anc 584 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
89883exp 1112 . . . . . . . 8 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9089adantr 481 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9190rexlimdv 3246 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦𝑋 (𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞))
9219, 91mpd 15 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
9314, 92eqtr4d 2834 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
949adantr 481 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
9511adantr 481 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
96 simpr 485 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
9795, 96fge0iccico 42194 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
9894, 97sge0reval 42196 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
9923, 27feqresmpt 6602 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
10099adantlr 711 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
101100oveq2d 7032 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))))
10221fveq2i 6541 . . . . . . . . . . 11 (+g𝐺) = (+g‘(ℝ*𝑠s (0[,]+∞)))
103 eqid 2795 . . . . . . . . . . . . . 14 (ℝ*𝑠s (0[,]+∞)) = (ℝ*𝑠s (0[,]+∞))
104 xrsadd 20244 . . . . . . . . . . . . . 14 +𝑒 = (+g‘ℝ*𝑠)
105103, 104ressplusg 16441 . . . . . . . . . . . . 13 ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞))))
10668, 105ax-mp 5 . . . . . . . . . . . 12 +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))
107106eqcomi 2804 . . . . . . . . . . 11 (+g‘(ℝ*𝑠s (0[,]+∞))) = +𝑒
108102, 107eqtr2i 2820 . . . . . . . . . 10 +𝑒 = (+g𝐺)
10921oveq1i 7026 . . . . . . . . . . 11 (𝐺s (0[,)+∞)) = ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞))
110 icossicc 12674 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ (0[,]+∞)
11168, 110pm3.2i 471 . . . . . . . . . . . 12 ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
112 ressabs 16392 . . . . . . . . . . . 12 (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞)))
113111, 112ax-mp 5 . . . . . . . . . . 11 ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
114109, 113eqtr2i 2820 . . . . . . . . . 10 (ℝ*𝑠s (0[,)+∞)) = (𝐺s (0[,)+∞))
11558elexi 3456 . . . . . . . . . . 11 𝐺 ∈ V
116115a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
117 simpr 485 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
118110a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
119 0xr 10534 . . . . . . . . . . . . 13 0 ∈ ℝ*
120119a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ∈ ℝ*)
12182a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → +∞ ∈ ℝ*)
12295ad2antrr 722 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
12326sselda 3889 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
124123adantll 710 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
125122, 124ffvelrnd 6717 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
12620, 125sseldi 3887 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ*)
127 iccgelb 12643 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑦))
128120, 121, 125, 127syl3anc 1364 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ≤ (𝐹𝑦))
129 id 22 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑦) = +∞ → (𝐹𝑦) = +∞)
130129eqcomd 2801 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) = +∞ → +∞ = (𝐹𝑦))
131130adantl 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ = (𝐹𝑦))
13211ffund 6386 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → Fun 𝐹)
133132ad2antrr 722 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → Fun 𝐹)
13422, 123sylan 580 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
13511fdmd 6391 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → dom 𝐹 = 𝑋)
136135eqcomd 2801 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 = dom 𝐹)
137136ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
138134, 137eleqtrd 2885 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
139 fvelrn 6709 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
140133, 138, 139syl2anc 584 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
141140adantr 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ ran 𝐹)
142131, 141eqeltrd 2883 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
143142adantl3r 746 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
14496ad3antrrr 726 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
145143, 144pm2.65da 813 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → ¬ (𝐹𝑦) = +∞)
146145neqned 2991 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
147 ge0xrre 41349 . . . . . . . . . . . . . 14 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
148125, 146, 147syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
149148ltpnfd 12366 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) < +∞)
150120, 121, 126, 128, 149elicod 12637 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
151 eqid 2795 . . . . . . . . . . 11 (𝑦𝑥 ↦ (𝐹𝑦)) = (𝑦𝑥 ↦ (𝐹𝑦))
152150, 151fmptd 6741 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)):𝑥⟶(0[,)+∞))
153 0e0icopnf 12696 . . . . . . . . . . 11 0 ∈ (0[,)+∞)
154153a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
155 eliccxr 12673 . . . . . . . . . . . 12 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
156 xaddid2 12485 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
157 xaddid1 12484 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
158156, 157jca 512 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
159155, 158syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
160159adantl 482 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
16172, 108, 114, 116, 117, 118, 152, 154, 160gsumress 17715 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
162 rege0subm 20283 . . . . . . . . . . . . 13 (0[,)+∞) ∈ (SubMnd‘ℂfld)
163162a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
164 eqid 2795 . . . . . . . . . . . 12 (ℂflds (0[,)+∞)) = (ℂflds (0[,)+∞))
165117, 163, 152, 164gsumsubm 17812 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
166 eqidd 2796 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
167 vex 3440 . . . . . . . . . . . . . 14 𝑥 ∈ V
168167mptex 6852 . . . . . . . . . . . . 13 (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V
169168a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V)
170 ovexd 7050 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂflds (0[,)+∞)) ∈ V)
171 ovexd 7050 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠s (0[,)+∞)) ∈ V)
172 rge0ssre 12694 . . . . . . . . . . . . . . . . 17 (0[,)+∞) ⊆ ℝ
173 ax-resscn 10440 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℂ
174172, 173sstri 3898 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℂ
175 cnfldbas 20231 . . . . . . . . . . . . . . . . 17 ℂ = (Base‘ℂfld)
176164, 175ressbas2 16384 . . . . . . . . . . . . . . . 16 ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂflds (0[,)+∞))))
177174, 176ax-mp 5 . . . . . . . . . . . . . . 15 (0[,)+∞) = (Base‘(ℂflds (0[,)+∞)))
178177eqcomi 2804 . . . . . . . . . . . . . 14 (Base‘(ℂflds (0[,)+∞))) = (0[,)+∞)
179110, 20sstri 3898 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ*
180 eqid 2795 . . . . . . . . . . . . . . . 16 (ℝ*𝑠s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
181180, 69ressbas2 16384 . . . . . . . . . . . . . . 15 ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞))))
182179, 181ax-mp 5 . . . . . . . . . . . . . 14 (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞)))
183178, 182eqtri 2819 . . . . . . . . . . . . 13 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞)))
184183a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞))))
185 rge0srg 20298 . . . . . . . . . . . . . . 15 (ℂflds (0[,)+∞)) ∈ SRing
186185a1i 11 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (ℂflds (0[,)+∞)) ∈ SRing)
187 simpl 483 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
188 simpr 485 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
189 eqid 2795 . . . . . . . . . . . . . . 15 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℂflds (0[,)+∞)))
190 eqid 2795 . . . . . . . . . . . . . . 15 (+g‘(ℂflds (0[,)+∞))) = (+g‘(ℂflds (0[,)+∞)))
191189, 190srgacl 18964 . . . . . . . . . . . . . 14 (((ℂflds (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
192186, 187, 188, 191syl3anc 1364 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
193192adantl 482 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
194172a1i 11 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
195 id 22 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
196195, 178syl6eleq 2893 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
197194, 196sseldd 3890 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ ℝ)
198197adantr 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ ℝ)
199172a1i 11 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
200 id 22 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
201200, 178syl6eleq 2893 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
202199, 201sseldd 3890 . . . . . . . . . . . . . . 15 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ ℝ)
203202adantl 482 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ ℝ)
204 rexadd 12475 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
205204eqcomd 2801 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
206162elexi 3456 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ∈ V
207 cnfldadd 20232 . . . . . . . . . . . . . . . . . . . . 21 + = (+g‘ℂfld)
208164, 207ressplusg 16441 . . . . . . . . . . . . . . . . . . . 20 ((0[,)+∞) ∈ V → + = (+g‘(ℂflds (0[,)+∞))))
209206, 208ax-mp 5 . . . . . . . . . . . . . . . . . . 19 + = (+g‘(ℂflds (0[,)+∞)))
210209, 207eqtr3i 2821 . . . . . . . . . . . . . . . . . 18 (+g‘(ℂflds (0[,)+∞))) = (+g‘ℂfld)
211210, 207eqtr4i 2822 . . . . . . . . . . . . . . . . 17 (+g‘(ℂflds (0[,)+∞))) = +
212211oveqi 7029 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
213212a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
214180, 104ressplusg 16441 . . . . . . . . . . . . . . . . . . 19 ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞))))
215206, 214ax-mp 5 . . . . . . . . . . . . . . . . . 18 +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞)))
216215eqcomi 2804 . . . . . . . . . . . . . . . . 17 (+g‘(ℝ*𝑠s (0[,)+∞))) = +𝑒
217216oveqi 7029 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
218217a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
219205, 213, 2183eqtr4d 2841 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
220198, 203, 219syl2anc 584 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
221220adantl 482 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
222 funmpt 6263 . . . . . . . . . . . . 13 Fun (𝑦𝑥 ↦ (𝐹𝑦))
223222a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦𝑥 ↦ (𝐹𝑦)))
224150, 177syl6eleq 2893 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
225224ralrimiva 3149 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
226151rnmptss 6749 . . . . . . . . . . . . 13 (∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
227225, 226syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
228169, 170, 171, 184, 193, 221, 223, 227gsumpropd2 17713 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
229165, 166, 2283eqtrd 2835 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
23030adantl 482 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
231148recnd 10515 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
232230, 231gsumfsum 20294 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
233229, 232eqtr3d 2833 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
234101, 161, 2333eqtrrd 2836 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (𝐺 Σg (𝐹𝑥)))
235234mpteq2dva 5055 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
236235rneqd 5690 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
237236supeq1d 8756 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
23898, 237eqtrd 2831 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
23993, 238pm2.61dan 809 . . 3 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
24021, 9, 11, 1xrge0tsms 23125 . . 3 (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
241239, 240eleq12d 2877 . 2 (𝜑 → ((Σ^𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )}))
2428, 241mpbird 258 1 (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1080   = wceq 1522  wcel 2081  wne 2984  wral 3105  wrex 3106  Vcvv 3437  cin 3858  wss 3859  𝒫 cpw 4453  {csn 4472   class class class wbr 4962  cmpt 5041  dom cdm 5443  ran crn 5444  cres 5445  Fun wfun 6219   Fn wfn 6220  wf 6221  cfv 6225  (class class class)co 7016  Fincfn 8357  supcsup 8750  cc 10381  cr 10382  0cc0 10383   + caddc 10386  +∞cpnf 10518  *cxr 10520   < clt 10521  cle 10522   +𝑒 cxad 12355  [,)cico 12590  [,]cicc 12591  Σcsu 14876  Basecbs 16312  s cress 16313  +gcplusg 16394   Σg cgsu 16543  *𝑠cxrs 16602  Mndcmnd 17733  SubMndcsubmnd 17773  CMndccmn 18633  SRingcsrg 18945  fldccnfld 20227   tsums ctsu 22417  Σ^csumge0 42186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-inf2 8950  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460  ax-pre-sup 10461  ax-addf 10462  ax-mulf 10463
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-fal 1535  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-int 4783  df-iun 4827  df-iin 4828  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-of 7267  df-om 7437  df-1st 7545  df-2nd 7546  df-supp 7682  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-fsupp 8680  df-fi 8721  df-sup 8752  df-inf 8753  df-oi 8820  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-div 11146  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-z 11830  df-dec 11948  df-uz 12094  df-q 12198  df-rp 12240  df-xadd 12358  df-ioo 12592  df-ioc 12593  df-ico 12594  df-icc 12595  df-fz 12743  df-fzo 12884  df-seq 13220  df-exp 13280  df-hash 13541  df-cj 14292  df-re 14293  df-im 14294  df-sqrt 14428  df-abs 14429  df-clim 14679  df-sum 14877  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-mulr 16408  df-starv 16409  df-tset 16413  df-ple 16414  df-ds 16416  df-unif 16417  df-rest 16525  df-topn 16526  df-0g 16544  df-gsum 16545  df-topgen 16546  df-ordt 16603  df-xrs 16604  df-mre 16686  df-mrc 16687  df-acs 16689  df-ps 17639  df-tsr 17640  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-submnd 17775  df-grp 17864  df-minusg 17865  df-mulg 17982  df-cntz 18188  df-cmn 18635  df-abl 18636  df-mgp 18930  df-ur 18942  df-srg 18946  df-ring 18989  df-cring 18990  df-fbas 20224  df-fg 20225  df-cnfld 20228  df-top 21186  df-topon 21203  df-topsp 21225  df-bases 21238  df-ntr 21312  df-nei 21390  df-cn 21519  df-haus 21607  df-fil 22138  df-fm 22230  df-flim 22231  df-flf 22232  df-tsms 22418  df-sumge0 42187
This theorem is referenced by: (None)
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